Proceedings of the **2004 **IEEE

**International Conference on ****Networking. Sensing & Control **
**Taipei, Taiwan, **March 21-23, 2004

**Iterative **

**ML **

**Estimation for Frequency Offset and **

**Time Synchronization in OFDM Systems **

**Chien-Chih **

**Chen**

Electrical Engineering Department National Taiwan University

Taipei, Taiwan 106, R.

### 0.

C. coolmail@tpts5 .seed.net.tw.

**Abstract **

### -

**In recent years, orthogonal frequency division**

**multiplexing (OFDM) technique has become attractive and**

**important in both wireline and wireless communications.**

**One of the well-known problems in OFDM systems is its**

**vulnerability to frequency and time offset. This paper**

**develops an iterative maximum-likelihood****(ML) estimation**

**algorithm for correction ofbothfrequency and time offset in**

**OFDM systems. Because of the complicated characteristics****of log-likelihood ****function, ****not every chosen initial point ****for ****iteration procedure can always converge to the ****values ****resulting in acquisition of the global maximum of log- ****likelihood function. This implies that some of the selected ****initial points can ****on!v converge to the values which yield ****local maxima of log-likelihood function. According to the ****proposed iterative approaching algorithm, the global ****maximum of log-likelihoodfunction can be definitely found ****to achieve both frequency and time synchronization. ****In **

**addition, the seeking time for this global maximum ****in ****the *** iteration procedure is able to be reduced and evaluated. *
Keywords: Maximum-likelihood estimation, frequency
offset, time synchronization, iterative approaching
algorithm, OFDM systems.

**1 Introduction **

Olthogonal frequency division multiplexing (OFDM)
is a powerful modulation scheme for high rate wireless
communication systems because of its superior property to
overcome frequency selective fading in multipath
environments. In Europe, OFDM has been standardized for
digital audio broadcasting (DAB) and digital video
broadcasting television [I]. Recently, OFDM has also been
applied to wireless local area network *(WLAN) ***such as **
IEEE 802.11a [Z] in United States and Hiperlad2 in
Europe.

In recent years, OFDM technique attracts increasing interests because of its excellent property in wireless communications. Since each subcarrier in OFDM is narrowband with respect to coherent bandwidth, an OFDM system is capable of coping with frequency-selective fading and narrow band noise. The other advantage in OFDM system is its efiicient modulation and demodulation schemes. Inverse discrete Fourier transform (IDFT) and discrete Fourier transform (DFT) can be implemented with inverse fast Fourier transform (IFFT) and fast Fourier

**Jung-Shan Lin **

Electrical Engineering Department National Chi

**Nan **

University
Nantou, Taiwan 545, R. ### 0.

C.j slin@ncnu.edu.tw

transform (FFT) which can be realized by specific hardware to speed up signal processing.

However, one of well-known problems in OFDM
system is its sensitivity to synchronization errors,
including both frequency and time. It is very sensitive to
frequency offset compared with other single-carrier
modulation schemes. All of the subcarriers in the OFDM
system are overlapped and orthogonal to each other. Minor
carrier frequency offset will make these subcarriers lose
**their orthogonality [3]. Like the sample time offset in time **
domain causing inter-symbol interference (ISI), frequency
offset will significantly degrade the system performance
and cause inter-carrier interference (ICI). For this reason,
how to detect carrier frequency and time offset is a crucial
problem in OFDM systems.

In general, frequency and time offset correction
**algorithms are classified into two folds based on using the **
additional data or not and what kind of additional data they
exploit. The data-aided schemes **are **suitable for the
applications that require fast and reliable synchronization
because of their redundancy of the OFDM data frame.
Training symbols (or called training sequence) and pilot
subcarriers are the **two **redundancy data that usually he
exploited. Training symbols are two or more consecutive
and identical symbols, and were used to estimate frequency
or **time offset in [3]. The other redundancy data, pilot **
**subcarrier, was first analyzed in single carrier system in [6J **
In multicarrier systems, the related synchronization
methods were proposed in **[7] **and **[SI. **The drawback of
data-aided method is the loss of data rate due to the
overhead.

The non-data-aided, or called blind, schemes relying
on the nature structure of OFDM frames to estimate
frequency offset. In [9] and [IO], the authors took
advantage of the presence of virtual carriers in OFDM
signaling to solve problem. Virtual subcarriers are the
carriers that are not modulated in order to avoid the
transmit filtering; hence, information-bearing subchannel is
smaller than the size of FFT block. How to select the null
subcarriers to result in better performance is discussed in
**[ I l l . Another blind method is first proposed in [I21 and **
[13], which were based on the correlation of the received
data samples. **An **adaptive algorithm is used to minimize
the mean square frequency offset error.

Figure **1. **An OFDM system structure

The other synchronization method for carrier-
frequency and time offset is to maximize the average log-
likelihood function [ **141 **with inserting the cyclic extension.
At the receiver side, the received signal with cyclic prefix
is used to obtain the joint likelihood function for
estimation of symbol frame and frequency offset. If the
maximum of ML function can be reached, then the actual
values of frequency and time offset can be simultaneously
obtained. Our work proposes an iterative approaching
algorithm to maximize the ML function and acquire the

time and frequency offset. A suggestion * is *also provided to
choose the starting point of iterative that can reduce the
trial-and-emor times of iteration and ensure to find the
correct global maximum

**of**likelihood function.

The remainder of this paper is organized as follows.
In Section 2, the basic shucture of an OFDM system
**model is introduced and analyzed. In Section 3, the **
analysis of the joint ML estimation for frequency and time
offset in the receiver end is presented and investigated.
The methodology of our iterative approaching algorithm
for the synchronization in both frequency and time is
derived and proposed in Section **4. ** Finally, some
concluding remarks are given in Section **5. **

**2 **

**OFDM **

**System **

**Model **

**A discrete-time FFTilFFT-based OFDM system is **
considered and its typical structure of block diagram is
shown in Figure 1. Each modulated symbol

*X, *

is
assumed to be an independent complex random variable.
Signal

**X k***is composed as N symbols*

*X, *

by inverse
discrete Fourier transform (IDFT), that is,
In wireless communications, inter symbol interference

**(1%) ** caused by multipath effect usually happens. To

eliminate this effect and maintain the orthogonality of OFDM signals, guard interval is usually inserted before

transmission. The guard interval, also called cyclic prefix

**(CP), **is a copy of last *L * * samples of xk appending to *
OFDM symbols as a preamble. After inserting the cyclic

**prefix, the transmitted frame becomes sk ,defined by****X N + ~ , ****k=O, **

### ...,

*L - l*

**x ~ - L , ** **k = L **

### ,....

*N + L - I '*(2)

* sk *= {

An OFDM **frame is depicted in detail in Figure 2. Finally, **
the parallel signals * sk are transferred to serial signal s ( k ) *
for transmission.
Frame

**k**### I

### 1

k d### j

' lorcnd**FFr wmdou**

**an OFDM blah**

Figure 2. The symbol h u e of OFDM systems.

**At the receiver side, r ( k ) is received and transferred *** to a parallel signal rk as a reverse of transmitter doing. *
Then, remove the first

*L*samples of frame

**rx **

, the relation
**rx**

**between yk and rk can be written as*** yk *=

**rk+L,****k **

= **k**

**0,**...,

*N*- 1

_{(3) }

**After that, through yk to the N-point discrete Fourier**transform (DFT) **as **follows:
**- j 2 h / N ****1 ** **N - l ****yn **=-

### c

**Yke***f i *

*(4)*

**n=O**If the orthogonal property of each subcarrier is preserved, the original data will be retrieved.

However, because of the non-stability of local
oscillators and Doppler effects, each subcarrier has a small
frequency offset mismatch between transmitter and
receiver. Besides, at the receiver side, there exists
uncertainty in symbol arrival time, **so the time offset is **
leaded. In discrete time system, frequency offset is
presented **as **a fraction of frequency spacing and denoted

**as **E , where

### I

E### I<

0.5. Assume all subcarrier experience the same offset, frequency offset can be modeled**as **

* e j2*” * in mathematically and multiply it to each

**s(k)**### .

As to another non-ideal effect, time offset, ‘is modeled as a

**step delay S ( k - 8 ) in the channel. In this****aticle,**transmitted signals

**are**assumed only suffer complex

**additive white Gaussian noise (AWGN)**

*variance*

**n(k) with***, the received signal*

**un**2**may**be written as

* r ( k ) *=

*-*

**s(k**

**8)e**jZnkE”### +

*(5)*

**n ( k ) .***enough, from central limit theory,*

**When the length of block N is**

**X k***can be seen as complex Gaussian*

**or sk**random variables with independent identical distribution
**(iid). And the sequence of transmitted signal, s(k) **

### ,

is acomplex Gaussian random process but not white due to
cyclic prefm insertion. The variance or the average energy
**of s ( k ) is u:, **

**3 ML Estimation Analysis **

**From the conclusion of [14], the log-likelihood **
function can be expressed **as **

A(@,&) =/

### 4

*(8)*

### I

C O S P E +*(8))*

### -

*PA2 (@),*

_{(6) }

where
B+L-l
_{(6) }

*Al(8)=*

**1 **

**r ( k ) r * ( k + N ) ,****k=e **

and
**k=e**

**(7) **

The magnitude of the correlation coefficient between

**rk **

**rk**

and

**rHN **

is
**rHN**

and

**SNR **

is defmed as *The estimated frequency*

**0****:****/c:.**offset

*8, *

**and time offset OML are the argument of 8 and*** E *that maximum this log-likelihood function, the answer
may be written as

**(B^ML,;ML)= **

a r g m a x A ( 8 , ~ ) . (10)
**(B^ML,;ML)=**

**( 0 6 ) **

The article [14] argues the maximum value happened when
* the cosine in ( 6 ) is equal to zero, then the closed form can *
be derived

**as**

(11)
* 8,* = argmax{~1(8)-pA2(8)]

**e **

and

In another point of view, * ( 6 ) is a function with two *
variables or axis, time and frequency. The function

**is**unlimited in time axis and is periodic in frequency axis because of the periodicity of cosine function, cos(&& + LAl (8)).

*question becomes finding a maximum point of (6) in the range of 8*

**If we concern only N+2L samples, the****E**

**[0 **

L] and E **E**[-0.5 0.51.

**4 **

**Iterative Approaching Algorithm **

In order to maximize the log-likelihood function
defmed in (6), utilizing iterative method with separation of
two variables is an appropriate and reasonable approach to
**this question. **In the log-likelihood function A ( ~ , E )

### ,

frequency offset*is continuous, but time offset 8 is discrete and only has fmite samples from 0*

**E**

**to N +****L- 1 .***Hence, choosing 0*

**as**the starting variable for the iteration procedure is better than starting at the

**axis**of frequency offset

**E**### .

### We

define*ei,j *

*whose initial seed is*

**is the im time-offset iteration result***eo,j *

at the time sample *j ,*

* j = 0 , - - - , N *+

*L - 1*

### .

That is to say,*is the fust choice we made to*

**eo,j****sfart**the iteration procedure. Then, (12) can be used to obtain the frequency-offset

**as**below:

where denotes the * i‘h * frequency-offset iteration result
whose initial seed is

*eo,j *

at time sample

**j**### .

Then, based on the known value of

**(6) becomes**### -

**(14)**

A(@)=l2,(@)

### I

cos(2n~,,j +*LA,(@))-*

### PA,(^.

Therefore, the log-likelihood function 4 8 , ~ ) with two variables becomes*x(8)*which depends only on

**0 . **

Then
the fEst iteration result is the argument that
maximizes the function *x ( S ) ,*and can be written as

should be substituted into (12) again and these steps
must he applied recursively, so we can obtain **~ 2 , ~ , ***B z r j , *

* E ~ , ~ , *and

*on. This iteration procedure continues*

**so**until the result of

**n **

-th time iteration is equal to that of
**n**

**(n**### +

**1)**-th time iteration, which can he expressed

**as**

As a result, the fmal iterative convergence pair (ICP),
written **as **(e;,&:), must be equal to * ( 8 n , j , ~ m , j ) , *that is,

**0;**=

*f f , , j*and

**s;**

*If the initial seed Bo,j is chosen*

**= E ~ , ~ .**appropriately, ICP will be the correct time and frequency offset we want to retrieve.

However, not all the initial seeds converge to the
correct values of frequency and time offset, and different
choices are possible to yield different results. Consider an
OFDM system with * N *=

**64 (number of the subcarriers)**and

**L =****16**(CP length) on an AWGN channel. Assume that the real frequency and time offset

**are**0.1 and 25, and

**SM****is equal to 25,The log-likeliiood function A(ff,&) of**the

**6" symbol frame is plotted in Figure 3. Clearly, we can**see that the global maximum occurs at the point

**( B = 2 5 , & = 0 . 1 ) , a n d t h e p o i n t ( B = 4 8 , & = - 0 . 3 2 ) i s a**local maximum. Using the previous mentioned iteration method directly without taking notice of the selection of initial seeds, the estimated values of time and frequency offset are possible to converge to local maximum points and retrieve incorrect results.

In **order to overcome this mistake, we can try every **
*initial point Bo,j and find a set of ICP *

### (e;,&;)

from*= 0 to*

**j***. Substihlte these pairs into equation*

**j =****N + L - I****(6), the value of**ML function with respect to each ICP can he evaluated and defmed by

* *

**(17) **
C . = * A(Bj,Ej) j = O...N+ L - *1.

*I *

From all **C j 's, ****choose the maximum value Cmax among **
them that is,

**The argument (ff;,.;) ** corresponding to ,C is
definitely the correct result for frequency and time offset.
Nevertheless, searching every possible point is tedious and
**wastes too much time. We have to find a way to reduce the **
seeking time for the actual values of frequency and time
offset if possible.

~

Figure **3. ** ML function surface, **6" frame, frequency **
offset==O.l,time offset=25,SNR=25. ~

Let us observe all the ICP values for every possible
**initial seed over a symbol frame in Figure 4. The seed is **
chosen from 0 to N + L - 1, which is **79 **in this case. The
actual time offset (TOS) and frequency offset (FOS) are
still 25 and 0.1, * S M = 2 5 , *and the observation frame is the

**6"symbol.**

*long as the initial seed is chosen at the*

**As****range**that the dotted and solid lines overlap, the proposed iteration method

**has**the ability to achieve the correct ICP.

In any frame * j *, the maximum successive length that the
dotted and solid lines overlap together is defmed as the
maximum iteration length

*d j .*in this case.

**Therefore, d6 is equal to 30**However, because of noise and the randomized
transmitted signal **s(k) **

### ,

the maximum iteration length is not always the same for evety frame symbol. Thisphenomenon can be observed in Figure 5. In this figure,
we plot the maximum iteration length of first **30 symhols **
with various subcarriers in an OFDM system. In general,
every symbol's maximum iteration length * d i *is usually
different. If we can fmd the exact iteration length

**d**### ,

which is defined as the smallest

**d j among all frames,**

**choosing initial seed once every d point, such as***ffo,j,B,,j+d,B,,jizd..., * and so forth, is enough to obtain
the actual values of frequency and time offset. According
to this rule, there must be at least one seed staying in the
region of maximum iteration length **of each frame, and **
therefore the seeking time can be reduced and evaluated in
the procedure.

**I **

**b **

**-04'**

**o ** i o **m ****30 ** **40 ** **m ** **a ** **70 ** **m **

**Figure 4. The distribution of correct convergence region at **
ream-

a symbol frame.

**fane i'dex **

Figure 5. The maximum iteration length for each frame. To estimate the exact iteration length

**d , **

we simulate
**d ,**

**OFDM system with different N and different****CP. The**minimum iteration length over 1000 frames is listed in Table 1 and plotted in Figure 6. It can be shown that these exact iteration lengths have almost linear relationship and all of them are larger

### than

or equal to the length of**CP,**

*which is L. As a result, choosing the exact iteration length*

*is a conservative assumption to guarantee acquisition of the global maximum of ML function, and then achievement of the synchronization of frequency and time.*

**d as L****Table 1: The exact iteration length over 1000 frames. **

**256 ** **49 ** **78 ** **101 ** **139 **

**512 ** **141 ** **I96 ** **251 **

Figure 6.The exact iteration length with various **CP lengths **
**To sum up, the proposed iterative approaching **
algorithm is summarized as following steps and the flow
chart of this algorithm is illustrated in the Figure 7.
* Siep 1: *Choose the initial seed to be

* Siep *2: Substitute tbe initial seed

*BOJ*into (13), and use (13)-(15) iteratively to find out the corresponding , and set

*= 0 .*

**j**.

**ICP **

### (s;,.;)

.* Step 3: *Substitute the

**ICP**

*into (17) to determine the value of log-likelihood function*

**( O ; , E ; )**

**Ci**### .

* Step4: *Increase

**j**by L (the length of**CP), that is,**

*j = *

*j + L . *

**If**

**the index j < N + L - l , then go**back to * Step 2. *Otherwise, go to

**Step****5.*** Step 5: *Determine the maximum from all the values of

*cj *

**'s**

**that are obtained in Step****3. The specific ICP**associated with this largest

*c j *

is the actual
answer for the values of frequency and time offset
**5 **

**5**

**Concluding **

**Remarks **

### In

this paper,### we

have developed the### maximum-

likelihood estimation with iterative approaching algorithm for correction of both frequency and time offset in OFDM systems. According to the proposed iterative approaching algorithm, the global maximum of log-likelihood function can be definitely found to achieve both frequency and time synchronization. In addition, we provided a method to reduce the seeking time for the global maximum of log-likelihood function in the iteration procedure.

Although this algorithm is only applied in the AWGN channel here, we believe that our iterative approaching algorithm can be extended to Rayleigb fading channels and multipath channels. As long as the ML functions

associated with various channel models can be analyzed and derived, our algorithm proposed here should have the potentials to maximize the log-likelihood functions and then achieve the synchronization in both carrier frequency and frame arrival time.

## .

? , I i .>### -

*I*

_{i }**t**

Figure **7. **The flow chart of the iterative approaching
algorithm.

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